\(\int (c+a^2 c x^2)^{5/2} \arctan (a x)^2 \, dx\) [326]

Optimal result

Integrand size = 21, antiderivative size = 516 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}} \]

[Out]

1/60*c*x*(a^2*c*x^2+c)^(3/2)-5/36*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a-1/15*(a^2*c*x^2+c)^(5/2)*arctan(a*x)/a+5
/24*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+1/6*x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2+259/360*c^(5/2)*arctanh(a*x*
c^(1/2)/(a^2*c*x^2+c)^(1/2))/a-5/8*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a
/(a^2*c*x^2+c)^(1/2)+5/8*I*c^3*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*
c*x^2+c)^(1/2)-5/8*I*c^3*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c
)^(1/2)-5/8*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+5/8*c^3*poly
log(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+17/180*c^2*x*(a^2*c*x^2+c)^(1/2)-
5/8*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a+5/16*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5000, 5010, 5008, 4266, 2611, 2320, 6724, 223, 212, 201} \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {a^2 c x^2+c}}+\frac {5}{16} c^2 x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {5 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{8 a}+\frac {5}{24} c x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {5 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{36 a}+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a}+\frac {17}{180} c^2 x \sqrt {a^2 c x^2+c}+\frac {1}{60} c x \left (a^2 c x^2+c\right )^{3/2} \]

[In]

Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

(17*c^2*x*Sqrt[c + a^2*c*x^2])/180 + (c*x*(c + a^2*c*x^2)^(3/2))/60 - (5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
(8*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(36*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(15*a) + (5*c^2*x
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/24 + (x*(c + a^2*c*x^2)^(
5/2)*ArcTan[a*x]^2)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c +
a^2*c*x^2]) + (259*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a) + (((5*I)/8)*c^3*Sqrt[1 + a^2*x
^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]
*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-
I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(
8*a*Sqrt[c + a^2*c*x^2])

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5000

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*p*(d + e*x^2)^
q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1))), x] + (Dist[2*d*(q/(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a +
b*ArcTan[c*x])^p, x], x] + Dist[b^2*d*p*((p - 1)/(2*q*(2*q + 1))), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])
^(p - 2), x], x] + Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

Rule 5008

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] &
& GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {1}{15} c \int \left (c+a^2 c x^2\right )^{3/2} \, dx+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \\ & = \frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {1}{20} c^2 \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{36} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {1}{40} c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{72} \left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{16} \left (5 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {1}{40} c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {1}{72} \left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {1}{8} \left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {c+a^2 c x^2}} \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{16 a \sqrt {c+a^2 c x^2}} \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = \frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.49 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (424 a x \sqrt {1+a^2 x^2}+368 a^3 x^3 \sqrt {1+a^2 x^2}-56 a^5 x^5 \sqrt {1+a^2 x^2}-11028 \sqrt {1+a^2 x^2} \arctan (a x)+504 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+12 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)+11970 a x \sqrt {1+a^2 x^2} \arctan (a x)^2+7380 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)^2+1170 a^5 x^5 \sqrt {1+a^2 x^2} \arctan (a x)^2-7200 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+8288 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+1550 \arctan (a x) \cos (3 \arctan (a x))+3210 a^2 x^2 \arctan (a x) \cos (3 \arctan (a x))+1770 a^4 x^4 \arctan (a x) \cos (3 \arctan (a x))+110 a^6 x^6 \arctan (a x) \cos (3 \arctan (a x))-90 \arctan (a x) \cos (5 \arctan (a x))-270 a^2 x^2 \arctan (a x) \cos (5 \arctan (a x))-270 a^4 x^4 \arctan (a x) \cos (5 \arctan (a x))-90 a^6 x^6 \arctan (a x) \cos (5 \arctan (a x))+7200 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-7200 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-7200 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+7200 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+372 \sin (3 \arctan (a x))+636 a^2 x^2 \sin (3 \arctan (a x))+156 a^4 x^4 \sin (3 \arctan (a x))-108 a^6 x^6 \sin (3 \arctan (a x))-1425 \arctan (a x)^2 \sin (3 \arctan (a x))-3555 a^2 x^2 \arctan (a x)^2 \sin (3 \arctan (a x))-2835 a^4 x^4 \arctan (a x)^2 \sin (3 \arctan (a x))-705 a^6 x^6 \arctan (a x)^2 \sin (3 \arctan (a x))-52 \sin (5 \arctan (a x))-156 a^2 x^2 \sin (5 \arctan (a x))-156 a^4 x^4 \sin (5 \arctan (a x))-52 a^6 x^6 \sin (5 \arctan (a x))+45 \arctan (a x)^2 \sin (5 \arctan (a x))+135 a^2 x^2 \arctan (a x)^2 \sin (5 \arctan (a x))+135 a^4 x^4 \arctan (a x)^2 \sin (5 \arctan (a x))+45 a^6 x^6 \arctan (a x)^2 \sin (5 \arctan (a x))\right )}{11520 a \sqrt {1+a^2 x^2}} \]

[In]

Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(424*a*x*Sqrt[1 + a^2*x^2] + 368*a^3*x^3*Sqrt[1 + a^2*x^2] - 56*a^5*x^5*Sqrt[1 + a^2*
x^2] - 11028*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 504*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 12*a^4*x^4*Sqrt[1 + a
^2*x^2]*ArcTan[a*x] + 11970*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 7380*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2
+ 1170*a^5*x^5*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (7200*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 8288*ArcT
anh[(a*x)/Sqrt[1 + a^2*x^2]] + 1550*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 3210*a^2*x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x
]] + 1770*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 110*a^6*x^6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 90*ArcTan[a*x]
*Cos[5*ArcTan[a*x]] - 270*a^2*x^2*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^4*x^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]]
- 90*a^6*x^6*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + (7200*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (7200*
I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 7200*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 7200*PolyLog[3, I*E
^(I*ArcTan[a*x])] + 372*Sin[3*ArcTan[a*x]] + 636*a^2*x^2*Sin[3*ArcTan[a*x]] + 156*a^4*x^4*Sin[3*ArcTan[a*x]] -
108*a^6*x^6*Sin[3*ArcTan[a*x]] - 1425*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3555*a^2*x^2*ArcTan[a*x]^2*Sin[3*Arc
Tan[a*x]] - 2835*a^4*x^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 705*a^6*x^6*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 52*
Sin[5*ArcTan[a*x]] - 156*a^2*x^2*Sin[5*ArcTan[a*x]] - 156*a^4*x^4*Sin[5*ArcTan[a*x]] - 52*a^6*x^6*Sin[5*ArcTan
[a*x]] + 45*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^2*x^2*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^4*x^4*ArcT
an[a*x]^2*Sin[5*ArcTan[a*x]] + 45*a^6*x^6*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]]))/(11520*a*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 a^{5} \arctan \left (a x \right )^{2} x^{5}-48 \arctan \left (a x \right ) a^{4} x^{4}+390 a^{3} \arctan \left (a x \right )^{2} x^{3}+12 a^{3} x^{3}-196 a^{2} \arctan \left (a x \right ) x^{2}+495 a \arctan \left (a x \right )^{2} x +80 a x -598 \arctan \left (a x \right )\right )}{720 a}+\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (225 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-225 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1036 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a \sqrt {a^{2} x^{2}+1}}\)	\(342\)

[In]

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/720*c^2/a*(c*(a*x-I)*(I+a*x))^(1/2)*(120*a^5*arctan(a*x)^2*x^5-48*arctan(a*x)*a^4*x^4+390*a^3*arctan(a*x)^2*
x^3+12*a^3*x^3-196*a^2*arctan(a*x)*x^2+495*a*arctan(a*x)^2*x+80*a*x-598*arctan(a*x))+1/720*I*c^2*(c*(a*x-I)*(I
+a*x))^(1/2)*(225*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-225*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^
2*x^2+1)^(1/2))+450*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*arctan(a*x)*polylog(2,I*(1+I*a*x
)/(a^2*x^2+1)^(1/2))+450*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(
1/2))-1036*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2)))/a/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^2, x)

Sympy [F]

\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**2,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)

Maxima [F]

\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^2, x)

Giac [F(-2)]

Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

int(atan(a*x)^2*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(atan(a*x)^2*(c + a^2*c*x^2)^(5/2), x)